On a generalization of a theorem of Erich Hecke

E. Hecke initiated the application of representation theory to the study of cusp forms. He showed that, for p a prime congruent to 3 mod 4, the difference of multiplicities of certain conjugate representations of SL(F_p) on cusp forms of degree 1, level p, and weight ≥2 is given by the class number h(-p) of the field Q(√-p). We apply the holomorphic Lefschetz theorem to actions on the Igusa compactification of the Siegel moduli space of degree 2 to compute the values of characters of the representations of Sp₄(F_p) on certain spaces of cusp forms of degree 2 and level p at parabolic elements of this group. Our results imply that here too, the difference in multiplicities of conjugate representations of Sp₄(F_p) is a multiple of h(-p).